Abstract
Interactions of quantum systems with their environment play a crucial role in resource-theoretic approaches to thermodynamics in the microscopic regime. Here, we analyze the possible state transitions in the presence of "small" heat baths of bounded dimension and energy. We show that for operations on quantum systems with fully degenerate Hamiltonian (noisy operations), all possible state transitions can be realized exactly with a bath that is of the same size as the system or smaller, which proves a quantum version of Horn's lemma as conjectured by Bengtsson and Zyczkowski. On the other hand, if the system's Hamiltonian is not fully degenerate (thermal operations), we show that some possible transitions can only be performed with a heat bath that is unbounded in size and energy, which is an instance of the third law of thermodynamics. In both cases, we prove that quantum operations yield an advantage over classical ones for any given finite heat bath, by allowing a larger and more physically realistic set of state transitions.
Highlights
The Superposition Principle (Physicists): If a quantum system can be in one of two mutually distinguishable states | A and | B, it can be both these states at once
If you look at the system, the chance of seeing it in state | A is |α|2 and in state | B is |β|2
Suppose we have a probabilistic quantum system which is in state vi with probability pi
Summary
The Superposition Principle (Physicists): If a quantum system can be in one of two mutually distinguishable states | A and | B , it can be both these states at once. It can be in the superposition of states α|A +β|B where α and β are both complex numbers and |α|2 + |β|2 = 1. The Superposition Principle (Mathematicians): The state of a quantum system is a unit vector in a complex Hilbert space If you have two qubits, their joint state space is the tensor product of their individual state spaces (e.g., C4)
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